Exact Barrier Option Valuation with Deterministic Volatility

Focus, in the past four decades, has been obtaining closed-form expressions for the noarbitrage prices and hedges of modified versions of the European options, allowing the dynamic of the underlying assets to have non-constant parameters. In this paper, we obtain a closed-form expression for the price and hedge of an up-and-out European barrier option, assuming that the volatility in the dynamic of the risky asset is an arbitrary deterministic function of time. Setting a constant volatility, the formulas recover the Black and Scholes results, which suggestsminimum computational effort. We introduce a novel concept of relative standard deviation for measuring the exposure of the practitioner to risk (enforced by a strategy). The notion that is found in the literature is different and looses the correct physical interpretation. The measure serves aiding the practitioner to adjust the number of rebalances during the option's lifetime.

在过去的四十年间，研究焦点集中于获取对欧式期权及其修改版本的零息价格和对冲策略的闭式解，并允许基础资产动态具有非恒定的参数。在本研究中，我们针对具有非恒定波动率的动态风险资产，推导出了一种上敲出欧式障碍期权的价格和对冲策略的闭式解。设定恒定波动率时，所得公式可还原为Black-Scholes模型的结果，从而表明了最小的计算工作量。我们引入了一种衡量从业者（受策略约束）风险暴露的相对标准差的新概念。文献中提出的概念与此不同，并丧失了正确的物理意义。该度量有助于从业者调整期权生命周期内的再平衡次数。